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2.5.8 Contemporary Physics 2

2.5.8.1 Overview

After the completion of Quantum Electrodynamics (QED) on the Electromagnetic Force, contemporary physics advanced to the Strong Force and the Weak Force employing matrices and Symmetric Theory.

2.5.8.2 Details

2.5.8.2.1 Hopper and Biswas' Discovery of Lambda Particles in 1950 CE

Background
Presence of various new particles was recognized.

Lambda Particles
Hopper and Biswas discovered a new particle, Lambda Particle (Λ particle). The Lambda particle had an interesting property changing into a proton and a π-meson. The mechanism was unknown at that time.
* "Lambda Baryon in Wikipedia" http://en.wikipedia.org/wiki/Lambda_baryon

2.5.8.2.2 Discoveries of New Particles through Particle Accelerators around 1952-1962 CE

Background
Particle accelerators were developed.

Discoveries
Various new particles were discovered employing particle accelerators.
For example, Xi particles (Ξ particles) were discovered in 1952 CE, Sigma particles (Σ particles) were discovered in 1953 CE.
* "Xi Baryon in Wikipedia" http://en.wikipedia.org/wiki/Xi_baryon
* "Sigma Baryon in Wikipedia" http://en.wikipedia.org/wiki/Sigma_baryon

2.5.8.2.3 Gell-Mann Nishijima Formula in 1953 CE

Background
Nambu suggested a partial theory on K mesons and Lambda particles (Λ particles) to his disciple, Nishijima.

Gell-Mann Nishijima Formula
Nishijima claimed a formula " Q = I3 + (B + S)/2 " classifying hadrons.
"Q" represents the electric charge of a particle (hadron).
"I3" represents the "Isospin 3-Component (the 3rd component of Isospin)" of a particle (hadron). Isospin has nothing to do with magnetic spin or spin quantum number, but Isospin's mathematical property is similar to spin's mathematical property.
For example, presuming "a proton and a neutron" to be analogues, isospin of 1/2 is given to "a proton and a neutron." Then the Isospin 3-Component of a proton is presumed +1/2, the Isospin 3-Component of a neutron is presumed -1/2.
Other than that, for example, isospin of 1 is given to Sigma particles. Then the Isospin 3-Component of Σ+ is presumed +1, the Isospin 3-Component of Σ0 is presumed 0, and the Isospin 3-Component of Σ- is presumed -1.
"B" represents the "Baryon Number" of a particle (hadron). Yet it merely means B of the Baryon (such as the proton, the neutron, and the Σ particle) is 1 and B of the Meson is 0.
"S" represents the "Strangeness Number" of a particle (hadron). Strangeness Number was a newly advocated quantum number to account for the strange property of Lambda particles (Λ particles) and K mesons.
Quantum numbers of the neutron are Q=0, spin=1/2, and B=1. On the other hand, quantum numbers of the Λ particle are equally Q=0, spin=1/2, and B=1. They couldn't be distinguished from a viewpoint of quantum numbers. Then a new quantum number, later "Strangeness Number," was advocated by Nishijima to enable to express the difference between neutrons and Λ particles. For example, Strangeness of the neutron is 0, Strangeness of the Λ particle is -1.
* "Gell-Mann Nishijima Formula in Wikipedia" http://en.wikipedia.org/wiki/Gell-Mann%E2%80%93Nishijima_formula
* "Isospin in Wikipedia" http://en.wikipedia.org/wiki/Isospin
* "Baryon Number in Wikipedia" http://en.wikipedia.org/wiki/Baryon_number
* "Strangeness in Wikipedia" http://en.wikipedia.org/wiki/Strangeness

2.5.8.2.4 Yang-Mills Symmetric Theory in 1954 CE

2.5.8.2.4.1 Background

Weyl's U(1) Gauge Symmetric Theory succeeded in theorizing the electromagnetic force. On the other hand, as mentioned before, Wigner presented a quantum number, Isospin, to distinguish protons and neutrons. For example, the Isospin of a proton is +1/2, the Isospin of a neutron is -1/2. Yang and Mills sought for a law that continuously links a proton and a neutron from a viewpoint of Isospin. (The mutual transformations between a proton and a neutron are mediated by the Weak Force. An example is Beta Decay when the numbers of protons and neutrons are unbalanced.)
* "Beta Decay in Wikipedia" http://en.wikipedia.org/wiki/Beta_decay
* "Proton-Neutron Ratio in Wikipedia" http://en.wikipedia.org/wiki/Proton%E2%80%93neutron_ratio
Yang and Mills examined U(1) Gauge Symmetry to link continuously. However, U(1) Gauge Symmetric Theory didn't fit in this case. Then improvement of U(1) Gauge Symmetric Theory was required.

2.5.8.2.4.2 Yang-Mills Gauge Symmetric Theory

Yang and Mills presented extended Gauge Symmetric Theory including SU(2) transformation, SU(3) transformation, and so on.
(Yang-Mills Gauge Symmetric Theory later brought about the Supersymmetric Theory.)
Their Gauge Symmetric Theory would be as follows.

Gauge Symmetric Theory could be inevitably difficultly defined as "theory involving continuous local transformations keeping Lagrangian constant." Gauge Symmetry would be the property featured by constancy of Lagrangian through such local transformations, consequently rotation transformations. When objects (particles) comply with Gauge Symmetry and objects (particles) are plotted depending on their obseved properties, they would be evenly distributed on the traces of a rotation such as the circumference of a circle and the spherical surface of a sphere. When objects (particles) wouldn't comply with Gauge Symmetry and objects (particles) are plotted depending on their observed properties, the distribution shown on the traces of a rotation such as the circumference of a circle would be biased.
* "Gauge Theory in Wikipedia" http://en.wikipedia.org/wiki/Gauge_theory
* "Gauge Symmetry in Wikipedia" http://en.wikipedia.org/wiki/Gauge_symmetry_(mathematics)

On the other hand, as mentioned before, it should be noted that such Gauge Symmetries (in a narrow sense to be precise) are not associated with realistic space or spacetime. Gauge Symmetries are employed on abstract mathematical conceptual things such as Magnetic Vector Potential. Since Gauge Symmetries have nothing to do with realistic space or spacetime, they are called "internal symmetry" in "internal spaces." "Internal spaces" are unrealistic abstract mathematical conceptual things. "Internal symmetries" are associated with bosons such as Photons, Weak Bosons, and Gluons.
* "Symmetry (physics) in Wikipedia" http://en.wikipedia.org/wiki/Symmetry_(physics)
In contrast, symmetries on realistic spacetime could be called "external symmetries" or "spacetime symmetries." They could be associated with the General Theory of Relativity or Fermions.

There could be some kinds of (internal) Gauge Symmetries, such as U(1) Gauge Symmetry, SU(2) Gauge Symmetry, and SU(3) Gauge Symmetry, consequently symmetries through rotation.
Consequently, SU(2) is symmetry evenly distributed on traces of rotations in (unrealistic conceptual) internal space consisting of 2 complex planes. SU(3) is symmetry evenly distributed on traces of rotations in (unrealistic conceptual) internal space consisting of 3 complex planes.

Since they are symmetries on traces of rotation transformations, the (conceptual) distance from the center point is constant, they are called "Gauge."
For example, "SU(2)" means "Special Unitary Transformation of 2x2 dimension." Unitary Transformation is commonly operated through "Unitary Matrices." For example, 2x2 corresponds to the group of 2x2 dimensioned "Unitary Matrices." "Special" in this case means "Determinant of the matrix is 1." SU means Special Unitary Group, a part of Unitary Group of matrices. These transformations could be called Gauge Transformations. (As mentioned before, matrices could be a kind of "Operators.")

In addition, "Gauge Symmetry" means the poroperty that "Lagrangian" is kept constant (invariant). "Lagrangian" is defined as " Lagrangian = T (kinetic energy) minus V (potential energy) = T - V ." (" V " here represents "potential energy" as formerly exemplified in the Shrodinger Equation.)
(In Newtonian mechanics, Energy of an object is defined as " Energy = T (kinetic energy) plus V (potential energy) = T + V " like " m*v^2/2 + V ." In contrast to that, " Lagrangian = T - V .")
As mentioned before, Lagrangian Mechanics employs 2 variables, " q " by "Generalized Coordinate System" and " " (" ● " crowned " q "), generalized velocity, as main variables to simplify (enable) mathematical analysis. (" q' " might be tentatively employed here instead of " ● " crowned " q " because of the difficulty of the letter type.) The ability of Lagrangian Mechanics and Lagrangian simplifying mathematical analysis through coordinate transformation implies essential perfectness of Lagrangian Mechanics.
As mentioned before, the physical meaning of "Lagrangian" would be "concentrated (or summarized) information on the motion." Then the constancy of Lagrangian under Gauge Transformations means "constancy (invariance) of essential physical property under Gauge Transformations (rotations)." Other than that, Lagranguan could be explained associated with the "Principle of Least Action" or "Brachistochrone Curve" of Johann Bernoulli, issues of, in a sense, economical (efficient) routes.
* "Unitary Operator in Wikipedia" http://en.wikipedia.org/wiki/Unitary_operator
* "Visual Physics U(1) Symmetry" http://visualphysics.org/de/qa/group-u1-electromagnetisms-gauge-symmetry
* "Special Unitary Group in Wikipedia" http://en.wikipedia.org/wiki/SU(2)
* "Lagrandian in Wikipedia" http://en.wikipedia.org/wiki/Lagrangian
* "Principle of Least Action in Wikipedia" http://en.wikipedia.org/wiki/Principle_of_least_action
* "Brachistochrone Curve in Wikipedia" http://en.wikipedia.org/wiki/Brachistochrone_curve

Other than that, according to Noether's Theorem, some physical property would be conserved along with a symmetry.
* "Conservation Laws and Symmetry in Wikipedia" http://en.wikipedia.org/wiki/Symmetry_(physics)#Conservation_laws_and_symmetry

2.5.8.2.4.3 Unitary Matrices

A Unitary Matrix or Unitary Matrices would be defined as " U U = I ," where U is a Unitary matrix (naturally a square matrix), U is "Conjugate Transpose Matrix" of U, "" is "dagger," a sign indicating conjugate transpose matrix, " I " here is "Identity Matrix."
* "Unitary Matrix in Wikipedia" http://en.wikipedia.org/wiki/Unitary_matrix
* "Conjugate Transpose in Wikipedia" http://en.wikipedia.org/wiki/Conjugate_transpose
As mentioned before, presuming the conjugate complex number and multiplying the original complex number and the conjugate complex number is the common way to calculate the size of the original complex number. On the other hand, according to the rules of Matrix Product, columns of the right (original) matrix correspond to rows of the left matrix. Then if the columns of the right matrix are changed into conjugate numbers and changed into the rows of the new left matrix, the original numbers of the columns could meet the brotherly conjugate numbers in the rows of the left new matrix through the multiplication. Transposing the columns to the rows results in adequate multiplication for each number.
This is the meaning of Conjugate Transpose Matrix.

U(2) Transformation
U(2) transformation is operated through U(2) matrix.
Prior to SU(2), the following is an example of U(2), a mere Unitary matrix (not Special Unitary matrix).



This is a Reflection Matrix.
The following shows a Reflection Matrix operated on a1 (4, 1).

1 +0*i 0 +0*i 4 4
=
0 +0*i -1 +0*i 1 -1

The definition of Gauge Symmetric Theory includes "continuous transformations ... ." However, reflection transformations wouldn't be continuous transformation. Specifically, the Gauge Theory requires Special Unitary Transformation (except U(1)), "Special" means determinant of the matrix = 1. However, the determinant of the above matrix = -1. Reflection matrices are not the matrices that the Gauge Symmetric Theory requires.

2.5.8.2.4.4 SU(2) Transformation

2.5.8.2.4.4.1 SU(2) Transformation on Real Numbers (Euclidean Space)

Rotation Matrices mentioned before are examples of SU(2) matrices (on real numbers).
(Space consisting of axes of real numbers is called Euclidean Space.)



In this case, θ = π/3, x = 4, and y = 1.
Then, the Rotation Matrix is

cos π/3 -sin π/3 4 cos π/3 * 4 - sin π/3 * 1
=
sin π/3 cos π/3 1 sin π/3 * 4 + cos π/3 * 1

Specifically,

0.5 -0.866 4 1.134
=
0.866 0.5 1 3.964

If the above matrix is called M,
M M = I .
Its determinant = cos(π/3)^2 + sin(π/3)^2 = 1 .
Then this is a SU(2) Matrix.
However, SU(2) Transformations generally range over complex numbers.

2.5.8.2.4.4.2 SU(2) Transformation on Complex Numbers (Hilbert Space)

Space consisting of axes of complex numbers is called Hilbert Space.
The following is an example of SU(2) Matrix.

cos π/3 i * sin π/3
i * sin π/3 cos π/3

If the above matrix is called M,
M M = I as

cos π/3 - i * sin π/3 cos π/3 i * sin π/3 1 0
=
- i * sin π/3 cos π/3 i * sin π/3 cos π/3 0 1

Its determinant = cos(π/3)^2 + sin(π/3)^2 = 1
Then, M is a SU(2) Matrix.

2.5.8.2.4.5 SU(3) Transformation

For example, a simple example of SU(3) Matrix is just extending the above SU(2) Matrix to 3x3 rotating around x-axis as follows.

1 0 0
0 cos π/3 i * sin π/3
0 i * sin π/3 cos π/3

Another simple example of SU(3) Matrix could be just rotating around z-axis counterclockwise by π/4 as follows.

cos π/4 - sin π/4 0
sin π/4 cos π/4 0
0 0 1

Then a more complex SU(3) Matrix would be synthesized multiplying these matrices, consequently as follows.

2 * SQRT(2)/4 - SQRT(2)/4 - i * SQRT(6)/4
2 * SQRT(2)/4 SQRT(2)/4 i * SQRT(6)/4
0 i * 2 * SQRT(3)/4 2/4


2.5.8.2.4.6 Non-Abelian Gauge Transformation

By the way, as mentioned before, U(1) Transformation is Abelian Gauge Transformation. In contrast, other Gauge transformations such as SU(3) Transformation and SU(2) Transformation are Non-Abelian Gauge Transformations.

For example, supposing 2 kinds of simple SU(3) operations, one example of Matrix operation is as follows. It rotates around x-axis counterclockwise by π/3. (Avoided imaginary numbers for simpler explanation.) It is tentatively called here " Mxπ/3 ."

1 0 0
0 cos π/3 - sin π/3
0 sin π/3 cos π/3

The other simple example of SU(3) operation here is rotating around z-axis counterclockwise by π/4 is as follows. (In this case, same as the previous example, though.) The following matrix is tentatively called here " Mzπ/3 ."

cos π/4 - sin π/4 0
sin π/4 cos π/4 0
0 0 1

In this case, the orders of operations are important.
For example, the coordinates of the original point is " x=0, y=0, z=4 ."
Case (a): firstly operate Mxπ/3 (rotate around x-axis counterclockwise by π/3), then secondly operate Mzπ/4 (rotate around z-axis counterclockwise by π/4).
The process is depicted as follows.
Firstly the original point (0, 0, 4) (the red dot) is rotated to the light purple dot (around x-axis), secondly the light purple dot is rotated to the dark purple dot (around z-axis).

Case (b): firstly operate Mzπ/4 (rotate around z-axis counterclockwise by π/4), then secondly operate Mxπ/3 (rotate around x-axis counterclockwise by π/3). The process is depicted as follows.
Firstly the original point (0, 0, 4) (the red dot) is rotated to the dark purple dot (around z-axis), while the dot stays at the same location (0, 0, 4) in this case. Then secondly the dark purple dot is rotated to the light purple dot (around x-axis).

Thus the results differ depending on the orders of operations. Then SU(3) Transformation is Non-Abelian, consequently Non-Abelian Gauge Transformation.
SU(2) is the same as well.
* "Non-Abelian Gauge Transformation in Wikipedia" http://en.wikipedia.org/wiki/Non-abelian_gauge_transformation

2.5.8.2.5 Discovery of Antineutrino and C-Symmetry Violation in 1956 CE

Background
Pauli predicted the presence of Neutrino in relation to the Weak Force relating to conversions between neutrons and protons (Beta Decay).
Cowan and Reines discovered Neutrinos in 1954 CE.
Yang and Mills claimed continuous Gauge Symmetry on Bosons and the significance of symmetries was implied.

Antineutrino
Cowan and Reines discovered Antineutrinos in 1956 CE. Consequently, Neutrinos' spins were all left-handed and Antineutrinos' spins were all right-handed spin. It means C-Symmetry is violated on Fermions of the Weak Force.
* "Neutrino in Wikipedia" http://en.wikipedia.org/wiki/Neutrino
*"Left-handed" and "C-Symmetry" are explained below.

2.5.8.2.6 Yang and Lee's Non-Conservation of Parity (P-Symmetry Violation) in 1956 CE

Background
Yang noted the significance of symmetries distributing evenly.
Gauge Symmetry was continuous symmetries. Discontinuous symmetries (discrete symmetries) such as reflection were yet unclear.
On the other hand K meson (Kaon) showed complicated decay.

Yang and Lee's Prediction on Non-Conservation of Parity
Yang and Lee claimed Non-Conservation of Parity (P-Symmetry Violation) during Kaon's decay.
* "NNDB Tsung-Dao Lee" http://www.nndb.com/people/790/000099493/
Subsequently, Wu verified Non-Conservation of Parity (P-Symmetry Violation) about the Weak Force in 1957 CE. According to Wu's experiment on the beta decay (Weak Interaction) of Cobalt-60, the directions of Beta emission were not balanced, it implied Non-Conservation of Parity (imbalance of Parity relationship) in the Weak Interaction.
* "Parity in Wikipedia" http://en.wikipedia.org/wiki/Parity_(physics)#Parity_violation
* "Hyperphysics Parity" http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/parity.html

*Parity (Parity Transformation, Parity Symmetry (P-Symmetry) or Parity Relationship) would be explained as follows. Parity Transformation would be a flip (reflection or inversion) like below on rather conceptual figurative things. The following is a depiction employing a Feynman Diagram. In the following example, the vertical direction designates time passage, the horizontal direction designates distance (space) like Minkowski Spacetime.

The left slanting line means the trace of a particle running from the past to the future and to the right.
The orange curved arrow of the left side means left-handed spin of the particle. As mentioned before, spins of particles are figurative conceptual mathematical unrealistic things to express some properties of particles. Actually, particles are not spinning. However, (assumed) angular momentum of spin and the direction of spin would be defined. The direction of (assumed) angular momentum of spin (the direction of spin) would be defined according to the right-hand screw.
Left-handed spin means that the direction of the spin and the direction of the particle (particle's momentum) (specifically for example, orbital angular momentum) are rather opposite. (Right-handed spin means that the direction of the spin and direction of the particle (particle's momentum) are similar.)
The parity relationship is depicted on the right like an image in a mirror. In this case, it should be noted that the direction of spin of the right side is consequently inverted, changed into right-handed spin.
Then Symmetry and its Violation are distinguished comparing each existing (observed) number of particles (comparing number of left-handed and number of right-handed in this case).
In addition, the property represented by "left-handed" and "right-handed" is called helicity.

*Other than that, C-Symmetry would be explained as follows. C-Symmetry means Charge Conjugation Symmetry. C-Symmetry would be similarly depicted as follows.

This is seemingly 180° rotation. However, since these are matters of discontinuous symmetries (discrete symmetries), C-Symmetry is theoretically the composite of P-Symmetry and T-Symmetry, to be precise.
T-Symmetry is depicted as follows.

T-Symmetry is time reversal symmetry inverting upside-down. In this case, the lower slant line is seemingly running from the future to the past. However, Feynman Diagram of this case represents an Anti-Particle running from the past to the future.
Other than that, the right-handed spin on the upper part of the T-Symmetry's depiction changed into left-handed spin (of the anti-particle).
Then referring to C-Symmetry, the C-Symmetry's depiction shows that the left-handed spin on the upper left changed into left-handed spin of the anti-particle on the lower right of C-Symmetry's depiction.
Consequently, C-Symmetry requires equal presence of left-handed Neutrinos and left-handed Antineutrinos. Left-handed Neutrinos were discovered, but left-handed Antineutrino is not discovered. Then, C-Symmetry on Neutrinos is violated.
*Such symmetry violation is exclusively on the Weak Force.

2.5.8.2.7 Landau's CP-Symmetry in 1957 CE

Background
C-Symmetry violation and P-Symmetry violation were verified on Weak Interaction (Weak Force), the mark (lighthouse) of physics became unclear.
Left-handed Neutrinos and right-handed Antineutrinos were discovered.

Landau's CP-Symmetry
Landau proposed CP-Symmetry as a new mark (lighthouse) instead of C-Symmetry and P-Symmetry. Since CP-Symmetry requires the balance between "left-handed particles and right-handed antiparticles" or "right-handed particles and left-handed antiparticles," the presence of left-handed Neutrinos and right-handed Antineutrinos comply with CP-symmetry.
* "CP Violation in Wikipedia" http://en.wikipedia.org/wiki/CP_violation
*CP-Symmetry is the complex of C-Symmetry and P-Symmetry. It corresponds to T-Symmetry. The T-Symmetry's depiction above merely shows "right-handed particles and left-handed antiparticles," but the other illustration left out here would be "left-handed particles and right-handed antiparticles."
However, as Wikipedia states CP Violation, violation of CP-Symmetry on the Weak Force was later discovered.
*CPT-Theorem states that CPT transformations result in invariance. Since C=P*T, CPT=P*T*P*T. These twice horizontal reflections (P) and these twice vertical reflections (T) naturally result in the original shape, invariance.

2.5.8.2.8 Everett's Many-Worlds Interpretation in 1957 CE

Background
Interpretation of Quantum Mechanics was controversial.

Everett's Many-Worlds Interpretaion
Everett claimed Many-World Interpretation. It claimed the presence of possible alternative histories and futures.
* "Many-Worlds Interpretation in Wikipedia" http://en.wikipedia.org/wiki/Many-worlds_interpretation
*It is referential in relation to J-Rod's Timeline story. However, it would be a mere fantasy fabricated by Yahweh.






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